A robust statistical estimation of the basic parameters of single stellar populations. I. Method

1.5 MB·English

by  X. Hernandez

Mon.Not.R.Astron.Soc.000,1–17(2007) Printed2February2008 (MNLATEXstylefilev2.2)
A robust statistical estimation of the basic parameters of single
stellar populations. I. Method
1 2
X. Hernandez and David Valls–Gabaud
1InstitutodeAstronom´ıa,UniversidadNacionalAuto´nomadeMe´xicoA.P.70–264,Me´xico04510D.F.,Me´xico
8
2GEPI,CNRSUMR8111,ObservatoiredeParis,5,PlaceJulesJanssen,92195MeudonCedex,France
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0
2
Accepted...Received...;inoriginalform...
n
a
J
ABSTRACT
9
Thecolour-magnitudediagramsofresolvedsinglestellarpopulations,suchasopenandglob-

ularclusters,haveprovidedthebestnaturallaboratoriestoteststellarevolutiontheory.Whilst
]
h a variety of techniqueshave been used to infer the basic propertiesof these simple popula-
p tions,systematicuncertaintiesarisefromthepurelygeometricaldegeneracyproducedbythe
- similarshapeofisochronesofdifferentagesandmetallicities.Herewe presentanobjective
o
r androbuststatistical techniquewhichlifts this degeneracyto a greatextentthroughthe use
t ofakeyobservable:thenumberofstarsalongtheisochrone.ThroughextensiveMonteCarlo
s
a simulationsweshowthat,forinstance,wecaninferthefourmainparameters(age,metallicity,
[ distanceandreddening)inanobjectiveway,alongwithrobustconfidenceintervalsandtheir

fullcovariancematrix.Weshowthatsystematicuncertaintiesduetofieldcontamination,unre-
2
solvedbinaries,initialorpresent-daystellarmassfunctionareeithernegligibleorwellunder
v
1 control.Thistechniqueprovides,forthefirsttime,aproperwaytoinferwithunprecedented
8 accuracy the fundamentalproperties of simple stellar populations, in an easy-to-implement
0 algorithm.
1
. Keywords: methods:statistical–stars:statistics–globularclusters:general–openclusters
1
andassociations:general–Galaxy:stellarcontent
0
8
0
:
v
1 INTRODUCTION theotherhand,reliesonthedifferenceinapparentmagnitudebe-
i
X tweenhorizontalbranch(HB)starsandturn-offstars.Bothmeth-
r Thetheoryofstellarevolutionhasbeentestedextensivelymainly odsrelyheavilyon“semi-empiricalcalibrations”providedbystel-
a larmodels,andaresubjecttomanyuncertainties(e.g.,Howtode-
throughwell-studiedbinarystars(e.g.,Popper1980,Lastennet&
fineproperlytheTOpointwhenphotometricerrorsblurthisregion
Valls-Gabaud 2002, Ribas 2006) or simple, coeval stellar popu-
ofthecolour-magnitudediagram?WhatistheverticaloffsetHB–
lationssuch asopen and globular clusters (e.g.Vandenberg et al.
TOwhentheHBisnothorizontal?)whichneverthelesshavebeen
1996,Renzini&FusiPecci1988).Ithasbeenextraordinarilysuc-
tackled withsome success (e.g.Rosenberg et al. 1999, Salaris&
cessful at explaining most stellar properties across the different
Weiss2002).
evolutionarystages,byincorporatingincreasinglycomplexphysics
in both stellar interiors and atmospheres (see Salaris & Cassisi Here we want to point out that many of the degeneracies
2006 for an updated and extensive review). In contrast, and per- areactuallymereartifacts,andthat aproper andrigorous mathe-
hapssomewhatparadoxically,thedeterminationofthefundamen- maticaltechniquecanbeformulatedtomeasuretheseparameters
talparametersofsinglestellarpopulations, suchasdistance,age, inarobust and objective way. For example, thewell-known age-
metallicity,reddening,convectiveparameters,etc,hasseldombeen metallicitydegeneracy,whichstatesthatagivenisochronecanbe
thesubject of arigorousmathematical determination. Moreoften fittedwithadifferentage providedthemetallicityischanged ap-
thannot,isochronesare“fitted”byeye,andveryroughestimates propriately, isa purelygeometricaldegeneracy : theshapeof the
oferrorsaregiven,ifgivenatall.Thereasonforthisstateofaffairs isochronesisthesame,yetstellarevolutionpredictsthatstarswith
isperhapsthemanydegeneraciesbetweentheseparameters,which a different metal content will evolve at a differentspeed. Hence,
makesitseeminglyimpossibletofindauniquesolutiontothesetof theage-metallicitydegeneracyisnotaphysicalone,butsimplya
fundamentalparameters.Alternative,semi-empiricalmethodshave resultofusingthegeometricalshapeofisochronesastheonlydis-
beendevelopedinsteadtoinferrelativequantities,ratherthanab- criminantfactor. Clearly,asimplestatisticalestimatorthatwould
soluteones.Forinstance,theso-calledhorizontalmethodrelieson countthenumberofstarsalongtheisochroneswouldgiveverydif-
the difference in colour between the turn-off (TO) point and the ferentresultsforanypairofage/metallicityvaluesthatwouldgive
baseoftheredgiantbranch(RGB),andisindependent oftheas- the same geometrical shape to the isochrones. In a more formal
sumptionsmadeontheconvectivetheory.Theverticalmethod,on way,letusconsideracurvilinearcoordinatesalonganisochrone
2 X. HernandezandD. Valls–Gabaud
ofgivenparameters(say,distance,reddening,metallicity,age,al- detailsalong the RGB.Clearly, theoptimal strategy isto use the
phaelementsenhancement,etc).Thispositiondependsonlyonthe fullinformationprovidedbytheobserveddensityofstarsalongthe
agetofthatisochroneandonthemassmofthestaratthatprecise isochrone.
locus,sothats=s(m,t)only.Anoffsetinagedtisreflectedonly
throughchangesinmassmandpositionsalongtheisochroneof Tosummarizeexistingmethods,oneapproachhasgrownout
agetsince of thefittingof ’geometrical’ indicators of theobserved CMD to
isochrones,fromsingleparticularindicatorssuchasturnoffpoint
∂t ∂t
dt(m,s) = dm + ds . (1) (TO), magnitude or colour difference between two points on the
∂m(cid:12)s ∂s(cid:12)m CMD, TO vs. zero-age horizontal branch or tip of the RGB and
(cid:12) (cid:12)
Forastarinthisis(cid:12)ochronetheoff(cid:12)setis,bydefinition,zero,hence RGB bump position and combinations thereof. Examples can be
∂m ∂m ∂s −1 found in Renzini & Fusi Pecci (1988), Buonanno et al. (1998),
= − × . (2) Salaris&Cassisi(1998), Salaris&Weiss(1998), VandenBerg&
∂s (cid:12)(cid:12)t ∂t (cid:12)(cid:12)s (cid:16)∂t(cid:12)(cid:12)m(cid:17) Durrel (1990), Ferraro et al. (1999) and Rosenberg et al. (1999).
The fi(cid:12)rst term is alw(cid:12)ays finite.(cid:12)The second term is the evolution- Theaccuracyoftheseapproacheshasbeenextendedbyincluding
aryspeed,therateofchangeforagivenmassmofitscoordinate several such geometrical indicators simultaneously (e.g. Vanden-
alongtheisochronewhentheagechangesbysomesmallamount. Berg 2000, Meissner & Weiss 2006), or a complete geometrical
One can thinkof s asrepresenting some evolutionary phase, and comparisonbetweenobservedCMDandisochrone(e.g.Straniero
so this term will be largewhen the phase is short-lived : a small & Chieffi 1991). The reduction of the CMD to a few numbers,
variation in age yields a very large change in position along the given the varied and complex physics which enters in determin-
isochrone.Alternatively,foragivenage,andsincethefirsttermis ingthemimpliesusingonlyasubsetoftheinformationavailablein
alwaysfinite,awidevariation inpositionimpliesanarrow range theCMD,whiletheunavoidableobservationalerrorspresentnec-
inmass.Thisisthecaseoftheredgiantbranchorthewhitedwarf essarilyleadtoambiguitiesintheidentificationofthekeypointsto
coolingsequence1,forinstance.Ontheotherhand,slowlyevolving beused, whicharehardtoquantifyformally, andwhichresultin
phasessuchasthemainsequencehavesmallevolutionaryspeeds uncertainconfidenceintervalsbeingassignedtotheinferencesde-
and wide ranges in mass for a given interval along an isochrone. rived.Neglectingstellarevolutionaryeffectslimitstheinformation
Clearly, themost important phases todiscriminatebetween alter- contentoftheisochronesagainstwhichtheCMDsarecompared.
nativeagesandmetallicitieswillbethepostmain-sequenceones,
wheretherangeofmassissmall(andhenceinsensitivetothede-
Ontheotherhand,approachesconcentratingexplicitlyonthe
tailsofthestellarmassfunction),andatthesametimewhereevo-
stellar evolutionary effects have grown out of the already men-
lutionaryspeeds arelarge.Ifweconsider themid-tolower main
tioned analysis of luminosity functions presented by Paczyn´ski
sequence,atfixedmetallicity,isochronesofallagestracethesame
(1984), a projection of the CMD onto the vertical axis. Exten-
locus, with only marginal changes in the density distribution of
sionshavemostlycenteredontheeasytomeasureandagesensitive
pointsamongstthem.Inthissense,oneoftheparameters,t,isto
featuresoftheRGBluminosityfunction(e.g.Jimenez&Padoan,
alargeextentabsentfromthemainsequence,whilephasesbeyond
1998,andZoccali&Piotto2000).Hereagain,observationalerrors
arealwayssubstantiallyafunctionofbothageandmetallicity.The
together withthe binning adopted in generating luminosity func-
densityofstarsalonganisochroneistherefore
tions, imply a limited use of the information available in the ob-
dN dN ∂m ∂t servation,whilenotincludingtheinformationencodedbythege-
= − × × . (3)
ds (cid:16)dm(cid:17) (cid:16) ∂t (cid:12)s(cid:17) (cid:16)∂s(cid:12)m(cid:17) ometryofboththeCMDandisochroneslimitstheaccuracyofthe
(cid:12) (cid:12) inferences.
Thefirsttermisrelatedtoth(cid:12)einitialstel(cid:12)larmassfunction(IMF),
and,asdiscussedabove,thesecondtermisfinitewhilethethirdisa
There are have been a few attempts in the past at using the
strongfunctionoftheevolutionaryspeed.Ifthemassaftertheturn-
full information available inacolour-magnitude diagram, as pio-
offisassumedtoberoughlyconstant,thisimpliesthattheratioin
neered by Flannery & Johnson (1982) and further developed by
the number of stars in two different evolutionary stages after the
Wilson & Hurrey (2003) and Naylor & Jeffries (2006). Unfortu-
turn-off will only depend on the ratio of their evolutionary time
nately all these approaches, while mathematically correct, fail to
scales2.
includeproperlythekeydiscriminatingfactordiscussedabove,the
Intermsofobservables,thiswell-knowrelationhasbeenex-
stellar evolutionary speed, resulting in wide uncertainties in the
ploited using luminosity functions as a proxy (as pioneered by
inferred parameters. The only other attempt we are aware of, in
Paczyn´ski 1984), that is, number counts of stars as a function of
thecontext ofinferringparametersfromastellarpopulation, was
apparent magnitude. This is not the isochrone coordinate s, but
madebyJørgensen&Lindegren(2005),whoadaptedourformal-
rather the projection of the isochrone on the vertical axis of the
ism developed in a previous paper (Hernandez, Valls-Gabaud &
colour-magnitudediagram.Thefactthatthesub-giantbranchoften
Gilmore1999)toderivestellarages.Thepresentpaperisafurther
appearsnearlyhorizontalinCMDsimpliesthatluminositybinsin
androbustextensionofthisapproach.PaperIIinthisseries(Valls-
this regime will not be discriminant. This is unfortunate because
Gabaud&Hernandez2007)willapplythemethodtodifferentsets
thisphaseismoreheavilypopulatedthantheredgiantbranch,and
ofobservationsofstellarpopulations.
thustheuseof luminosityfunctionsisnotoptimal. Likewise,the
projection onthehorizontal axis, acolour, islesssensitivetothe
In section 2 we present the rigorous probabilistic model,
whichistestedwithsyntheticcasesinSection3.Insection4weex-
ploreandquantifytheeffectsofsystematicuncertainties,insection
1 excludingthebottomofthecoolingsequence,wherewhitedwarfshaving
5wepresentanapplicationofthemethodtoarealcase,NGC3201.
alargerangeofprogenitormassespileup
2 Inthecontextofstellarpopulation synthesis,thisisknownasthefuel Finallyinsection 6wepresent our conclusions on thisnovel ap-
consumptiontheorem(Renzini&Buzzoni1983). proach.
Robuststatisticalmeasures ofsinglestellarpopulations 3
2 PROBABILISTICMODEL where σ(L )2 and σ(C )2 are the errors which have to be ap-
m m
pliedtothetheoreticalcomparisonstarofmassmtomapinonto
Intryingtorecoverthephysicalparametersofanobservedcluster,
theobservedCMD.i.e.,eachtheoreticalstarisbeingthoughtofas
webeginbyconstructingamodelwhereonly4numbersdetermine
aGaussianprobabilitydensitydistributioncenteredon(L ,C ),
theabove,theage,metallicity,distanceandreddeningoftheclus- m m
withdispersionsσ(L )andσ(C ).Equation4ishenceequiva-
ter,avector(T,Z,D,R).Henceforth,wewilldefineDasthetotal m m
lenttotheconvolutionovertheentireCMDplaneoftwoGaussian
verticaldisplacementofthestarsintheobservedCMDwithrespect
probabilitydistributions,afirstonecomingfromthereportedob-
totheir original positions.Thisverticaldisplacement isofcourse
servedstar,andasecondonefromtheprobabilistictransferfunc-
thedistancemodulusandthecontributionoftheextinctioninthe
tionwhichwemustapplytoatheoreticalmodelstarcomingfrom
observedband,thatis,D = µ+R × R,whereRisthecolour
V anisochronetomapitontoaparticularobservedCMD.
excessinthebandsconsideredandR theratioofselectivetototal
V Ifwewantedtoconsidermodelstarsasmathematicalpointsin
absorption.Notethatthesecontributionsare,ingeneral,correlated,
thiscomparison,followingBayestheorem,wewouldidentifythe
sinceatlargerdistancesthereddeningusuallyincreaseswithinthe
probabilitythataparticularreportedrealstar,itselfa2-Dprobabil-
Galaxy.NotealsothatR maydependonthelineofsightconsid-
V itydistribution,wasinfactacertainmodelstar,withtheprobabil-
ered,eventhoughauniversalextinctioncurveisusuallyassumed.
itythatthisparticularmodelstarmightactuallycomefromthe2-D
Forthesereasonsitissimpler(andmodel-independent)toconsider
probability distribution of the reported star. To treat the problem
a vertical offset, and a horizontal one, the colour excess, as two
rigorously, we consider only the comparison between a reported
independentquantities.
starandamodelone,onlyafterthislastonehasbeenmappedonto
Thus,weassumetheIMFtobeknown,thebinaryfractionto
theplanewheretheobservedstarexists,throughaprobabilityfunc-
be zero and the contamination of background/foreground starsto
tion,asdescribedabove.
benegligible.Thisconditionsareofcoursenottrueinarealcase,
InEquation4anumericalnormalizationconstanthasbeenleft
andwewillthereforetreatthemassystematics,theeffectsofwhich
out,asinanycaselikelihood functionshavenoabsolute normal-
weshallexploreindetailinsection4.
izations and only relative values are meaningful. Whereas σ(L )
FromaBayesianperspective,givenanactualobservedCMD, i
andσ(C )aresuppliedbyaparticularobservationofarealCMD,
we want to identify the vector (T,Z,D,R) which has the high- i
σ(L )andσ(C )havetobeestimated.Inapracticalapplication,
est probability in having resulted in the actual observed cluster, m m
onecouldobtaintheaverageofσ(L )andσ(C ),asfunctionsof
as that from which the CMD most probably actually came from. i i
magnitude,andusethosetwofunctionsasmodelsforσ(L )and
Thismodel isessentially what was introduced and tested inHer- m
σ(C ).
nandez et al. (1999), and subsequently used in Hernandez et al. m
Havingtheprobabilitythatasingleobservedstarisinfacta
(2000a,2000b)toinferstarformationhistoriesofresolvedgalax-
particularmodelstar,wecannowcomputetheprobabilitythatour
ies,althoughmuchsimplifiedhereforthetreatmentofsinglestellar
observedstarcamefromanypointalongaparticularisochronein
populations. Wemustconstruct astatisticalmethodfordetermin-
ourparameterspace(T,Z,D,R)as:
ingwhatistheprobabilitythatanobservedCMDmighthavebeen
theresultofagivenvector (T,Z,D,R).AnobservedCMDwill
m1(T,Z)
be defined by a set of i = 1 ··· N stars, each having a mea-
⋆ G (T,Z,D,R)= ρ(m;T,Z)G (T,Z,D,R,m)dm.(6)
sured magnitude and colour (in whichever photometric bands or i Z i
filters) (Li,Ci), with σ(Li) and σ(Ci) being the measured (de- m0(T,Z)
ducedfromphotometrydatareductionprocedures)1-sigmaerrors
In the above expression ρ(m;T,Z) is the density of stars
in these quantities. Errors in magnitudes and colours will be as-
alongtheisochroneofmetallicityZandageT,aroundtheteststar
sumedasbeingGaussianinnature,anduncorrelatedtoeachother
ofmassm.Asexplainedin§1,thisdensitydependsbothontheas-
forthetimebeing.Hence,areportedstar(L ,C ;σ(L ),σ(C )),
i i i i
sumedIMFandontheevolutionaryspeedpredictedbythestellar
willbethoughtofasatwo-dimensionalGaussianprobabilityden-
evolutioncodeatthatpositionoftheCMD.Thedensityofpoints
sitydistribution,centeredon(L ,C ),withdispersionsσ(L ),and
i i i
alongtheisochrone,withinacertainintervalaroundacertainmass,
σ(C ).
i
inthehighlydiscriminatingregionbeyondthemainsequence,will
We begin with the probability that a given observed star, of
be determined essentially by the time spent by a star within that
magnitudeandcolour(L ,C ),isinfactarealstarofsomeaget,
i i
interval,asgivenbythestellarevolutionarycodes.Ifalargepor-
andsomemetallicity,Z,ofmassm,luminosityL andcolourC ,
m m
tion of the main sequence is included in the study, the IMF will
observedwithanassumedverticaloffsetofD,andareddeningof
dominateρ(m;T,Z) overthatregion, asdiscussedin§1. Onthe
R,bothinunitsofmagnitudes.Thiswillbegivenby
otherhand,themassintervalbeyondtheturnoffpointissosmall,
thatpracticallyanyconceivableIMFwillyieldidenticalresultsfor
Gi(T,Z,D,R,m) = ρ(m;T,Z)inthisregion.TheRBGhasalowerdensityofpoints
thanthemainsequencenotbecauseoftheIMF,butbecauseofthe
1
× faststellarevolutionaryspeeds.
(cid:18)σL(m,i)σC(m,i)(cid:19)
In practice, the extremely rapid variations in magnitude and
× exp −DL2(m,i) ×exp −DC2(m,i) . (4) colour with mass over critical regions of the isochrones make it
(cid:18) 2σ2(m,i) (cid:19) (cid:18) 2σ2(m,i) (cid:19) convenient to carry out the integration in Equation 6 over a pa-
L C
rameter measuring length over the isochrone, rather than mass,
IntheaboveexpressionD (m,i)andD (m,i)arethedif-
L C in which case ρ(m;T,Z) must include weighting factors due to
ferencesinmagnitudeandcolourbetweentheithobservedstarand
the relative duration of different stellar phases. Also, m (T,Z)
agenericstarofage,metallicity,distancemodulus,reddeningand 0
and m (T,Z) are a minimum and a maximum mass considered
mass(T,Z,D,R;m).Also, 1
alongeachisochrone,suchthattheluminositycompletenesslimit
σ2(m,i)=σ(L )2+σ(L )2 , σ2(m,i)=σ(C )2+σ(C )2,(5) of the observations always corresponds to the luminosity of star
L i m C i m
4 X. HernandezandD. Valls–Gabaud
m (T,Z), while m (T,Z) corresponds to the star of age and whichsomeoftheoffspringsineachgenerationhavearandomly
0 1
metallicity (T,Z) at the tip of the RGB. The explicit exclusion selected“gene”switchedfrom“1”to“0”,orvice-versa.Thissim-
of phases beyond thehelium flashreduces to acertain extent the plisticalgorithmhasbeenshowntoconvergeremarkablyefficiently
resolutionofthemethod,butincreasessignificantlyitsrobustness, intotheglobalmaximumofcomplexmulti-dimensional surfaces,
asitguarantees (seesection4) thatonly well-establishedphysics associatedtoverydifferentproblems.Amorecompletedescription
areconsideredinthestellarevolutionarymodels. ofthealgorithmcanbefoundinCharbonneau(1995),withsuccess-
We shall refer to G (T,Z,D,R) as the likelihood matrix, fulapplicationstodiverseastrophysicalproblemsbeingdescribed
i
since each element represents the probability that a given star, i, ine.g.Teriacaet.al(1999) forfittingradiativetransfermodelsto
wasactuallyformedwithmetallicityZ,attimeT withanymass, solaratmospheredata,Sevensteretal.(1999)infittingparameters
and that itwas observed witha distance modulus of D, and red- ofGalacticstructuremodelstoobservations,Metcalfe(2003),for
deningofR.Thatis,inthisBayesianapproach,wetreatthestellar thefittingofwhitedwarfastroseismologyparametersorGeorgiev
massmasanuisanceparameterwhichwemarginalizeover. &Hernandez(2005)intheproblemofinferringoptimalwindpa-
The probability that an entire observed CMD with N stars rametersfromobservedspectrallineprofiles,amongmanyothers.
⋆
arosefromaparticularmodel,i.e.,fromaparticularchoiceofpa-
rameters(T,Z,D,R),willnowbegivenby:
N⋆ 3 TESTINGTHEMETHOD
L(T,Z,D,R) = G (T,Z,D,R) . (7)
i InthissectionwetestthemethoddescribedinSection§2,through
Yi=1 theinversionofaseriesofsyntheticCMDs.Asinthiscasethepoint
In this way, we have constructed an optimal merit function inparameterspacefromwhichaparticularCMDwassimulatedis
which can be evaluated at any point within the relevant four di- known,wecanoptimallyassestheperformanceofthemethod.
mensionalparameterspace(T,Z,D,R),givenanobservedCMD. Insimulatinganobserved CMDthefirststepistoobtainan
Itisimportanttoremarkthatthismeritfunctionusesallinforma- adequateisochronelibrary.Differentgroupshavesucceededinpro-
tionavailableintheCMD densely. Nobinning or averagingover ducingstate-of-the-artstellarevolutionarycodesusingmostly,but
discreteregionsisnecessary,andnoaspectsofthedistributionof not all the time, the same input physics. Since systematic errors
stars on the CMD are discarded. Also, all the rich stellar evolu- will be produced by the choice of the set of isochrones, we use
tionaryphysicsisincludedinthecomparisonwiththemodels,as theresultsofthethreedifferentgroups(Bergbusch&VandenBerg
theisochrones arenotbeing reducedtoadiscreteset ofnumbers 1997, VandenBerg et al. 2006, Demarque et al. 2004, Girardi et
(e.g.thepositionoftheTO,RGBslope,etc.)orevencurvesonthe al.2002).Wehaveworkeddirectlywiththeoutputofstellarevo-
CMD.Asweshallseeinsection§3,thefulluseofallfeaturesin lutionary codes to produce our own isochrone grid with a dense
both data and models, allows aprecise estimation of the inferred grid including 180 time steps and 120 metallicity intervals. Each
parameters. isochronecontainsapproximately300stellarmasses.Thetimeres-
Following Bayes theorem, one would now want to identify olutionforagesofbetween4.5Gyrandtheupperlimitof18.0Gyr
thatpointwhichmaximizesL,the4coordinateswhichmaximize isof0.2Gyr,withyoungeragesbeingsampledmoredensely.The
theprobabilitythattheobservedCMDdidinfactarisefromapar- spacinginmetallicityislinearinthelogarithm,with0.03dexper
ticularpointintheparameterspace,withthesetofphysicalcluster interval. Throughout the paper we shall identify metallicity with
parameterswhichhavethehighestprobabilityofhavinggivenrise [Fe/H], when using theoretical isochrones, or comparing against
totheobservedCMD. empirical inferences. The spacings of the isochrone grid will de-
Severalmethodsexistforidentifyingtheglobalmaximumofa terminethemaximumresolutionofthemethod,independently of
multidimensionalsurface.Moststandardmethodsrunintoconsid- whatapproachistakenatthefollowinglevelsoftheproblem.For
erabledifficultiesincaseswherelocalmaximaexist,asourprob- thepurposes of presenting themethod, these gridsare more than
lemturnsouttohave.Themostreliablewayofsolvingthisproblem appropriate.
istoevaluateL throughoutadensegridinalloftheavailablepa- Itisimportanttonotethatthehighlynon-linearvariationsin
rameterspace.Thishowever,ishighlytimeconsuming,andhence magnitudeandcolourwithmasswhichappearalongisochroneson
itishighlydesirabletofindamoreefficientmethod. criticalregionsoftheCMDimpliesthatadirectinterpolationbe-
Wehaveusedthegeneticalgorithmapproach,whereanumber tweentwoisochroneswillnotproducetheintermediaryone,butan
of“agents”areplacedatrandompointswithinourparameterspace, averagecurvewhichbareslittleresemblancetotheoutputofastel-
andthepositionofeachcodifiedintoabinarynumber,the“genes” larevolutionarycode.Inallcases,wehavelimitedtheisochronesto
ofeachagent.Thenumberofbinarybitsusedforeachcoordinate belowtheheliumflash.Thisconditionrestrictstheinformationwe
fixestheresolutionofthealgorithm,whichinthiscasecorresponds considerinourmethodtotheregionwheredisagreementbetween
totheresolutionoftheisochronegridonthe(T,Z)plane,andto differentgroupsworkinginthefieldisataminimum.
0.01maginthe(D,R)plane.ThevalueofListhenevaluatedat Oncetheisochroneshavebeenestablished,anIMFischosen
thepositionofeachagent,andtheresultisusedasafitnessfunc- todistributestarsamongstdifferentmasses.Thisfunctionprovesto
tion.Asecondgenerationisthenconstructedby“mating”theorig- berelativelyunimportant,asthemassrangeofstarsoneisdealing
inalagents,pairingthemandcombiningthe“genes”ofeachparent withissmallprovidedstarsareselectedfromonemagnitudeorso
toproducetwo“offsprings”.Thefitnessfunctionisusedtoweight belowtheTOregionallthewaytothetipoftheRGB.Thisensures
thematingprobability,suchthatthefitterindividualshaveagreater that the mass range probed by the observations is small. For ex-
probabilityof“mating”.The“genes”oftheoffspringsareacombi- ample,inglobularclustersolderthan∼5Gyrthemassintervalis
nationofthoseoftheirparents,andhenceasgenerationsproceed, around1M⊙andhenceinthisnarrowmassrangethedetailsofthe
thepopulationgraduallymovesintoregionswherethefitnessfunc- IMFareirrelevant.Thedistributionofstarsalongtheisochroneis
tionishigher,i.e.,intoregionsofhighL.Toavoidgettingtrapped amuchmoresensitivefunctionofthestellarevolutionarymodels,
intolocalmaxima,aslight“mutationrate”isintroduced,through whichisinfactwhatdeterminesthatthemainsequenceishighly
Robuststatisticalmeasures ofsinglestellarpopulations 5
13 13.5 14 14.5 -3 -2.5 0.05 0.1
0.1
-2.6 14.5
-2.8
0.08
14
-3
0.06
-3.2 13.5
-3.4
0.04 13
18.9 18.9 18.9
18.8 18.8 18.8
18.7 18.7 18.7
13 13.5 14 14.5 -3 -2.5 0.05 0.1
T(Gyr) log(Z) E(V-I)
Figure1. (Left)SimulatedSingleStellarPopulationCMDhavinganageof13.9GyrandlogZ=−2.7.Thethick(black)curveshowstheinputisochrone,
withtheoptimalfitisochronerecoveredbythemethodappearingasathin(red)curve.Dottedcurvesshowtheyoungestandoldestisochronesallowedbythe
method,atthe1−σlevel,whichinfactspana1.0Gyrinterval.(Right) ThesixpanelsshowtheprojectionoftheerrorellipsoidresultingfromtheMonte
CarlosimulationandinversionofaSSPwithage13.9GyrandmetallicitylogZ=−2.7,ontodifferentplanes.Thefilledcirclesshowinputparameters,and
theemptyonestheaveragevaluesfortherecoveredparameters.
populated,theredgiantbranchmuchmoresparselyso,andthatthe functionofthesimulatedCMDwithrespecttoanygivenpointon
HRgapregionwillcontainveryfewstars.TheIMFweuseisthat our 4-Dparameter space. Asdiscussed insection §2,theseeking
of Kroupa et al. (1993), although as mentioned above, any other oftheoptimalpointwillbecarriedoutthroughageneticalgorithm
approximation to this function would yield the same results (see simulation, the workings of which, for this first example, we de-
§4.1). scribebelow.
HavinganisochronegridandanIMF,wecannowpickapoint The upper left panel of Figure 2 shows the evolution of the
inour(T,Z,D,R)parameterspaceandplaceachosennumberof valueofthelikelihoodmeritfunctionof Equation7over thefirst
starsintothemagnitude-colourplane.Forthefirstimplementation 100generationsofthegeneticsimulation.Thehorizontallinegives
we have simulated an extreme example with close to 2000 stars thevalueofthissamefunction,evaluatedattheactualpointinour
around point (13.9,−2.7,18.8,0.05), a single stellar population (T,Z,D,R)parameterspacewhichwasusedtoproducethesyn-
of age 13.9 Gyr, metallicity −2.7 (that is, close to −1.0 in solar theticCMDbeinginverted.Thisfirstpanelshowsanincreasewith
units,for asolar metallicityof logZ⊙ = −1.7), withadistance thegenerationnumberoftheoptimalvalueofthelikelihoodfunc-
modulus of 18.8 magnitudes, and observed through a reddening tion, as the absolute best point found in all the simulation up to
of 0.05 magnitudes. Next we model magnitude and colour errors a given generation is what is being plotted. This value increases
as Gaussian with sigmas as appropriate for current HST or high veryrapidlyatfirst,astheinitiallyrandompointsbegintomigrate
qualitylongintegrationground-based observations. Explicitlywe towardsregionsmorecloselyresemblingthesimulatedCMD,but
used progress becomes increasingly slower. In fact, there is no defini-
tivecriteriontoestablishwhenageneticalgorithmsimulationhas
σ(L)=0.001×exp[(L+D−17.0)/2.25]
finallyconverged,thishastobeinferredfromsimulationswithsyn-
theticdataandrepeatedexperiments,asanapproximatenumberof
σ(C)=0.001×exp[(L+D−17.0)/1.97]
generations, suitableforaparticular problem, afterwhichnofur-
whereListhemagnitude,intheV band,ofthetheoreticalstar,and therimprovementisachieved.Fortheproblembeingtreatedhere,
C itsV −I colour.Atthispointwecannowproduceasynthetic wehavefoundthemethodstopsyieldinganyimprovementinthe
CMD,andforthisfirsttestobtainthe2009pointsshowninFigure fitsin general well before 3500 generations. Wehave thus ended
1. A magnitude cut at V = 25 has been used, limiting analysis thegeneticsimulationsat3500generations.Thevalueofthelike-
tostarsbrighter thanthiscutoff.Giventhehighly correlatedway lihoodfunctionevaluatedattheactualinputpointalwaysremains
inwhich star populate the main sequence, together withgrowing abovetheonecorrespondingtotheoptimalonefoundbythesimu-
and sometimes divergent errors towards higher magnitudes, it is lationforthisfirst100generationsshownhere.
sufficienttorestrictanalysisto2magnitudesbelow the’turnoff’ The upper right panel of Figure 2 gives the evolution of the
region.Inwhatfollowsweshallkeepthe25magnitudelowerlimit, optimaltimegridindex(TI)alongtheisochronegrid,foundupto
although achanging magnitude cut asdescribed above yieldsthe acertainpoint.Inthiscase,theinputageof13.9Gyrcorresponds
sameresults. toour timeindex 159 out of 180, over thisinitial100 generation
Atthispoint,Equation7canbeusedtoobtainthelikelihood period the time index corresponding to the optimal model found
6 X. HernandezandD. Valls–Gabaud
0 20 40 60 80 1000 20 40 60 80 100 0 1000 2000 3000 0 1000 2000 3000
150 150
100 100
0
50 50
0 0
100 1 100 1
80 80
60 0 60 0
40 40
20 -1 20 -1
0 0
0 20 40 60 80 1000 20 40 60 80 100 0 1000 2000 3000 0 1000 2000 3000
Generation Generation Generation Generation
Figure2. (Left) Thefourpanelsshow,inaclockwiseprogressionstartingattheupperleft,themaximumlikelihoodpointfoundbythemethod,theoptimal
timeindex,theoptimaldistancemodulus,andtheoptimalmetallicityindex,allasfunctionsofgenerationnumberforthegeneticalgorithmsimulation,forthe
first100generations.(Right) Sameastheleftpanels,butshowingthefull3500generationsofthegeneticalgorithmsimulation.
fluctuates from an initial value of 120 to 178, from 6 to 18 Gyr, onetimegridintervalabove(0.2Gyr),11metallicitygridintervals
andendsthisinitialperiod12gridpointsabovetheinputvalue,at below(0.4dex)andatadistancemodulusaround0.05magnitudes
16.9Gyr.Thebottomleftandrightpanelsshowthecorresponding largerthantheinputparameters.
evolutionofthemetallicityindexandthedistancemodulus,again, Theaboveresultmakesitclearthatinferringstructuralparam-
thehorizontallinegivestheinputparameters.Wecanseethatafter etersofstellarpopulations,eveninthesimplestcases,cannotbe
2generations theinitiallyrandom valueshaveapproached signif- attemptedwithoutafullconsiderationoftheprobabilisticnatureof
icantly the input ones, and then show considerable variations of theproblem,andalso,cannotgiveerror-freeresults.However, it
around 35 metallicity grid intervals (1.5 dex) and 0.8 mag in the ispossibletoevaluatetheintrinsicerrorsofthemethod,inconnec-
distancemodulus. tiontoanyparticularimplementation,bytheMonteCarlomethod.
Itisinterestingtore-plottheleftpanelof Figure2thistime Theexperiment isrepeated alargenumber of times,andthevar-
overtheentire3500generationrange,thisisdoneintherightpanel ious answers analyzed to obtain mean inferred values with well
ofFigure2wherewecanseethatsubstantialimprovementsinthe determineddispersionsandcorrelations.
optimal likelihoodvaluefound arestilloccurring after1000 gen- This method naturally takes into consideration all internal
erations.Astrikingfeatureseenintheupperleftpanelofthislast sources of error and dispersion inherent to the procedure being
Figureisthatthegeneticalgorithmhasactuallyfoundanoptimal tested,andyieldsaccurateandreliableconfidenceintervalsonthe
model which in fact yields a likelihood value above that of the inferences obtained. Figure1 (right) shows with filledcirclesthe
input parameters. At around 1000 generations, theoptimal likeli- input parameters of the CMD of Figure 1 and with empty ones
hoodfoundcrossesthehorizontallinegivingthelikelihoodvalue thecenterofthedistributionsoftherecoveredparametersafterthe
ofEquation7oftheinputparameters,withrespecttothisparticu- wholeprocedureofconstructingasyntheticCMDandinvertingit
larrealizationoftheCMDcorrespondingtothem.Thisapparently was repeated 20 times. The ellipses give projections of the error
paradoxicalresultisinfactunavoidable,andhighlightsthestatisti- ellipsoidontothedifferentplanesbeingshown.Itcanbeseenthat
calnatureoftheproblem.TheconstructionofaCMDfromapoint theonlysystematicwhichappearsinthiscasebeyondthe1σlevel
inthesinglestellarpopulationparameterspacecannotbethought isan+0.03magnitudeoffsetintherecoveredvaluesoftheredden-
ofasneitheradeterministicnorauniqueprocess.Evenbeforethe ingforthecluster.Allotherparametershavebeenrecoveredwith
errorsassociatedwithprocuringtheobservationsenterthepicture, nosystematics,the1σerrorsforthis13.9Gyroldclusterbeing0.5
the process of sampling an IMF has already introduced a proba- Gyr in age, 0.38 dex in metallicity and 0.05 mag in the distance
bilisticingredient intrinsictothecluster itself(e.g.Cervin˜oetal. modulus.
2002,Cervin˜o&Valls-Gabaud2003).Inthissense,obtainingasin- Theisochronecorresponding tothesolidcirclesisshownin
glesimulatedCMDandusingittocomparewithanobservedone Figure1bythethicksolidcurve,whichpracticallyliesabovethe
canonlybeatbestacrudefirstapproximationtotheinvertingofa thin(red)curveoftheisochronecorrespondingtotheemptycircles
CMD.Inthisparticularcase,theconvolvingofthetheoreticalstars alongalltheCMD.Somedifferencesbetweenthetwoisochrones
witherrorgaussianshasshiftedthepointsaroundinsuchawaythat are evident along the upper half of the red giant branch, where
theisochrone(understoodasaseriesofstellarevolutionarymod- so few stars are expected, that none of the 2009 obtained in the
els,notjustacurveonaCMD)whichhasthegreatestprobability CMD realization shown appear there. The dotted curves give the
ofhavingyieldedtheCMDisnottheinputone,butonewhichis isochronescorrespondingtotheoldestandyoungestpointsonthe
Robuststatisticalmeasures ofsinglestellarpopulations 7
11 12 -3.4 -3.2 -3 0.04 0.06
-3 0.07
12.5
-3.1
0.06
12
-3.2
0.05 11.5
-3.3
11
0.04
-3.4
10.5
-3.5 0.03
18.85 18.85 18.85
18.8 18.8 18.8
18.75 18.75 18.75
18.7 18.7 18.7
11 12 -3.4 -3.2 -3 0.04 0.06
T(Gyr) log(Z) E(V-I)
Figure3. (Left)SimulatedSingleStellarPopulationCMDhavinganageof10.9GyrandlogZ=−3.05.Thethick(black)curveshowstheinputisochrone,
withtheoptimalfitisochronerecoveredbythemethodappearingasathin(red)curve.Dottedcurvesshowtheyoungestandoldestisochronesallowedbythe
method,atthe1σlevel,whichinfactspana1.48Gyrinterval.(Right) ThesixpanelsshowtheprojectionoftheerrorellipsoidresultingfromtheMonte
Carlosimulationandinversionofasinglestellarpopulationwithage10.9GyrandmetallicitylogZ =−3.05,ontodifferentplanes.Thefilledcirclesshow
inputparameters,andtheemptyonestheaveragevaluesfortherecoveredparameters.
ellipseintheage-metallicityplane, whichareseentobracket the anextensiveregionofthe(T,Z)plane,merelyarelationbetween
HR-gapandRGBregionsoftheCMDshown. thepossibleageandmetallicity,withnowayofrestrictingthean-
The second example was constructed from point swertoanarrowerregion.Theabovehighlightsproblemsinherent
(10.9,−3.05,18.8,0.05) in our parameter space, taken to to methods where information is being discarded, to which must
represent atypical old SingleStellarPopulation (SSP).Intesting beadded ambiguitiesassociated tothedefinitionof the’turnoff’
themethodforsystematicsrelatedtoeffectsnotexplicitlyincluded point.Giventhecurrenthighaccuracyoftheobservationsandthe
inthemodel,comparisonswillbemadeatthispointinparameter theoreticalisochrones,inspectionofsimulatedandobservedCMDs
space,insection§4. makesitclearthatthereisinfactnosuchthingasaturnoff’point’,
Figure3showstheCMDwhichresultsfromconvolving the butratheradiffuse,hazyregionwhichispronetosamplingfluctu-
modeledstarswiththesamesimulatederrorsthanlasttime.Also ationsandthepresence,orotherwise,ofafewoddstars.
shownaretheisochronefromwhichthesimulatedstarswerecon- Other correlations are evident in Figures1 and 3, for exam-
structed,thick(black)curve,andtheoptimalonefoundbythege- pleaclearnegativecorrelationbetweentheinferredreddeningand
neticalgorithm,thincontinuouscurve.Thedottedcurvesgivethe metallicity,alsoagenericfeatureofanystellarpopulationinversion
oldest and youngest isochrones consistent with the Monte Carlo method,inherenttothestructureoftheisochrones.Itisinteresting
simulationontheinvertingprocedureata1σlevel,thistimespan- tonote inthelower 3panels of Figures1 and 3that the inferred
ning1.2Gyr.AnalogoustoFigure1,Figure3(right)givesthepro- distancetotheclustershowsnostrongcorrelationswithanyofthe
jectionoftheerrorellipsoidfortheinferredparametersusingthe other3parameters,contrarytowhathappenswhentheisochrones
MonteCarlomethod,projectedonto6differentplanes.Oneagain are reduced to a set of numbers such as location of the turn off
seesthattherearenosystematicsbeyondthe1σlevel,fromwhich point,slopeoftheRGBetc.
weconcludethatthemethodaccuratelyrecoverstheinputparam- Forathirdexample, intryingtocover points onour param-
etersofthecluster,andyieldsmeaningfulconfidenceintervalsfor eterspacerepresentativeofoldSSPs,wesimulatedanold,metal
theanswersupplied. poorsuchsystem.Startingfrompoint(13.9,−4.0,18.8,0.05)we
LookingattheupperleftpanelofFigure3(right)weclearly construct a series of CMDs, one of which is shown in Figure 4,
see the actual intrinsic age-metallicity degeneracy in the way the withcurvesanalogoustowhatappearsinFigures2and3.Itisim-
1σ ellipse in the (T,Z) plane slants from the upper left to the pressivetoseehownearlyindistinguishabletheshapesoftheinput
lowerright.Optimalfitswithlargeagestendtocorrespondtolower andmeanrecoveredisochronesareontheCMD,togetherwiththe
metallicities,andvice-versa.However,weseethathavingincluded isochronescorrespondingtotheoldestandyoungestisochroneal-
alltheinformationavailablebothinthesimulatedCMDandinthe lowedbytheMonteCarlosimulationata1σ level.All4ofthese
theoreticalisochrones minimizesthisdegeneracy andoneobtains curves appear practically on top of each other in this case, with
closedconfidenceintervals.IftheCMDisreducedto,saytwonum- veryminordifferencesonlyapparentatthecurvaturebetweenthe
bers,correspondingtothelocationofthe’turnoff’,whencompar- HRgapregionandtheRGB.
ingtoasimilarlyrestrictiveview ofwhat thestellarevolutionary Theaboveisparticularlysurprisingwhenconsideringthatthe
modelsencode,oneendsupwithaconfidenceregionwhichspans isochrones shown span almost a 1.5 Gyr-long time interval. Of
8 X. HernandezandD. Valls–Gabaud
12.5 1313.5 14 -4 -3.5 0.04 0.06
-3.2 0.07
14
-3.4
0.06
13.5
-3.6
0.05
-3.8 13
0.04
-4
12.5
-4.2 0.03
18.85 18.85 18.85
18.8 18.8 18.8
18.75 18.75 18.75
18.7 18.7 18.7
12.5 1313.5 14 -4 -3.5 0.04 0.06
T(Gyr) log(Z) E(V-I)
Figure4. (Left) SimulatedSingleStellarPopulation(SSP)CMDhavinganageof13.9GyrandlogZ = −4.0.Thethick(black)curveshowstheinput
isochrone,withtheoptimalfitisochronerecoveredbythemethodappearingasathin(red)curve.Dottedcurvesshowtheyoungestandoldestisochrones
allowedbythemethod,atthe1σlevel,whichinfactspana1.44Gyrinterval.(Right) Thesixpanelsshowtheprojectionoftheerrorellipsoidresulting
fromtheMonteCarlosimulationandinversionofaSSPwithage13.9GyrandmetallicitylogZ=−4.0,ontodifferentplanes.Thefilledcirclesshowinput
parameters,andtheemptyonestheaveragevaluesfortherecoveredparameters.
course, the other parameters besides the age have been adjusted stellar populations the mean recovered parameters appear within
tomaximizethefitwhenshiftingtheage,inaccordance withthe onegridintervalinmetallicityoftheinputparameters,thetheoret-
correlations shown inFigure 4, theoldest isochrone has thelow- icalmaximumresolutionoftheimplementationhasbeenrealized
estmetallicitytogetherwithasomewhataboveaveragereddening attheseyoungages,evenafterhavingconsideredrealisticerrorsin
correction. Thispoint again showsthat theisochrones havetobe magnitudesandcolours.Theageresolutionisstillabovethatofthe
treatedascollectionsoffullstellarevolutionmodels,notreduced isochrones,butrobust1σerrorsofaround0.5Gyrareencouraging.
toafewscalarshapeparameters,noteventosimplyshapesonthe Weaddthattheprecisionofthemethodinrecoveringdistance
CMD. Havingused thefull shape of theisochrones continuously modulusandreddeningparametersissufficienttomakeitauseful
throughoutthefullCMD,togetherwithinformationregardingthe toolinthedeterminationofdistancestosinglestellarpopulations,
durationofdifferentstellarphasesinconstructingthemeritfunc- witherrorssometimesasgoodasafewhundredthsofamagnitude,
tionofEquation7hasallowedamuchfinerdiscriminationwithin as seen in Figure 5. Again, the correlations persist, although the
our(T,Z,D,R)parameterspace,muchbeyondwhatconsidering details of these correlations change from example to example in
onlytheshapeoftheisochroneswouldhaveallowed.Changingthe waysthatreflectthedetailsoftheisochronesaroundeachregion,
inputparametersoftheisochroneevenata2-3sigmalevel,would and which are dependent on both the magnitude of the assumed
result in curves compatible with the shape of the simulated stel- observationalerrorsandtheactualnumbersofstarspresentinthe
lardistribution,butwhichareruledoutduetohavingincludedall simulatedCMDs.
availableinformationbothintheCMDandthestellarmodels. With simulated errors tending to zero the inferred parame-
Again,nosystematicoffsetsbetweentheinputvaluesandthe ters would tend to the input ones, to within the resolution of the
meanrecoveredonesappearsbeyondthe1σ level,ascanbeseen isochronegridused,always.Also,ifthestatisticalcomparisonbe-
from Figure 4. In this same Figure we see the same correlations tween data and models is robust, the systematics and confidence
evidentinthepreviousanalogousFigures,withtheconfidencein- intervalsshouldtendtozeroasthenumberofdatapointstendsto
tervals,particularlyinageandmetallicityhavinggrownconsider- infinity,whichofcourseisneverthecase.
ablyincomparisontoFigure3.Thisisobviouswhenoneconsid-
ershow consecutive isochrones equally spaced intimewillshow
progressivelysmallerdifferencesastheageofastellarpopulation
increases.Havinglefttheassumedobservationalerrorsandstellar
4 SYSTEMATICS
numbersconstant,theuncertaintiesintheinferredparametershave
growningoingtoanold,metalpoorpopulation. Havingtestedsuccessfullythemethodacrossrepresentativepoints
Forthefourthandfifthexampleswecompleteoursamplingof inthe(T,Z,D,R)space,wenowturntotheproblemofestimating
SSPsparameterspacewithtwoclustersat(6.9,−2.0,18.8,0.05) howrobusttheinferencesofourmethodwillbeinmorerealistic
and(6.9,−3.37,18.8,0.05).Figures5and6arecompletelyanalo- cases, where physical ingredients not explicitlyconsidered in the
goustothecorrespondingpreviousfiguresandessentiallyillustrate statistical modeling behind Equation 7, and present in a real ap-
the same points mentioned before. For these younger simulated plication, will complicate matters, and might result in systematic
Robuststatisticalmeasures ofsinglestellarpopulations 9
6.5 7 -2.1 -2 -1.9 0.02 0.04 0.06
-1.9 0.07 7.4
0.06 7.2
-1.95
0.05 7
-2
0.04 6.8
-2.05
0.03 6.6
-2.1 0.02 6.4
18.85 18.85 18.85
18.8 18.8 18.8
18.75 18.75 18.75
18.7 18.7 18.7
6.5 7 -2.1 -2 -1.9 0.02 0.04 0.06
T(Gyr) log(Z) E(V-I)
Figure5. (Left)SimulatedSingleStellarPopulationCMDhavinganageof6.9GyrandlogZ=−2.05.Thethick(black)curveshowstheinputisochrone,
withtheoptimalfitisochronerecoveredbythemethodappearingasathin(red)curve.Dottedcurvesshowtheyoungestandoldestisochronesallowedbythe
method,atthe1σlevel,whichinfactspana1.0Gyrinterval.(Right)ThesixpanelsshowtheprojectionoftheerrorellipsoidresultingfromtheMonteCarlo
simulationandinversionofaSSPwithage6.9GyrandmetallicitylogZ =−2.05,ontodifferentplanes.Thefilledcirclesshowinputparameters,andthe
emptyonestheaveragevaluesfortherecoveredparameters.
offsetsbetweentheactualphysicalparametersofastellarpopula- oftheMonteCarloprocedurecanbeseeninFigure7,whichfrom
tionandourinferences. the previous discussions would not have been expected to show
Inthissectionweexploresuchcasesbyconstructingsynthetic any substantial difference to Figure 3, as is indeed the case. By
CMDsunderassumptionsdifferenttotheonesthemethodusesin comparingwithFigure3wenotenosubstantialdifferences,allin-
theinversionprocedure,andcomparehowserioustheseeffectsare ferredparametersarewellwithinthe1σerrorestimates,withvery
inalteringourinferences.The5effectsconsideredwillbe:(1)Er- lowuncertaintiesofaround0.5Gyrinage,0.2dexinmetallicity,
rorsintheassumedIMF,(2) thepresenceofunresolved binaries, 0.01and0.05maginreddeninganddistancemodulus,respectively.
(3)theeffectsofreducingthenumbersofstarspresentintheob- Theisochronescorrespondingtotheinput,inferred,andoldestand
servedCMD,(4)systematicsassociatedwiththedifferentsetsof youngestvaluesrecovered(ata1σlevel),practicallyoverlapover
stellartracksusedinconstructingtheisochronegrids,and(5)the theentireregionoverwhichsignificantnumbersofstarsarefound.
presenceofcontaminatingstarsintheobservedCMDs. Weseethatthetheoreticalexpectationsoftheprevioussectionsare
borne out, and our results are in fact highly robust to even large
scalefeaturesoftheIMF.Thisisreassuringfromthepointofview
4.1 EffectsoftheIMF
ofinferringages,metallicities,distancesandreddeningcorrections
Figure7givesaCMDresultingfromthesamepointinparameter ofresolvedstellarpopulations,butdishearteningfromthatofinfer-
spaceasFigure3,(10.9,−3.05,18.8,0.05),butcreatedthistime ringtheIMFofobservedstellarclustersthroughmethodssimilar
usingasubstantiallydifferentIMF,heavilyweightedtowardsmas- towhathasbeendevelopedhere.
sive stars, increasing the slope with respect to the Kroupa et al.
(1993)IMFsuchthatthetypicalstellarmassincreasesbyafactor
4.2 Effectsofunresolvedbinarystars
of3.Obtainingthesamenumberofpointsthusrequiresturninga
much largertotal massintostars, butunder theassumption of an Thesecondtestwasperformedagainusingthesameisochroneli-
equalnumberofstarsasinFigure3,theresultsarehighlysimilar. brary, and the input point (10.9,−3.05,18.8,0.05) of Figure 3,
Theprecisemanner inwhichthedensityofstarsvariesalongthe butthistimetooneintwoofthesimulatedstarsweaddedanun-
mainsequenceregionissomewhatdifferentfromthecaseofFig- resolvedcompanion,chosenfromthesameIMF,i.e.,thesynthetic
ure3,butgiventhehighlycorrelatedmannerinwhichmagnitudes CMDwasproducedassuminga50%unresolvedbinarycontribu-
andcoloursscalealongthisregion,theinformationcontentofthe tion.Theinversionprocedurewashoweverexactlythesameasin
CMDisonlymarginallyaltered.Thehighlydeterminant sections thepreviouscases,withEquation7remainingunchanged,abinary
beyond the main sequence span such a small mass range that all fractionof0%wasassumedinrecoveringtheinferredparameters.
ofthesefeaturesarevirtuallyleftunchangedbyeventhestrongest OneoftheresultingCMDswhichwasusedintheMonteCarlo
modificationsintheIMF. procedure for the inversion is shown in Figure 8, together with
Theinversionprocedurewasrepeated20times,butassuming isochronesasinthepreviousFigures.Thepresenceof50%bina-
thesameIMF asinFigure3, whichinthiscasenolonger corre- riesisparticularlyconspicuousalongthemainsequence,wherethe
spondedtotheIMFusedtocreatethesyntheticCMDs.Theresults distributionofpointsisclearlysmearedtowardsreddercolours,ev-
10 X. HernandezandD. Valls–Gabaud
6.5 7 7.5 -3.6 -3.4 -3.2 0.02 0.04 0.06
-3.2 0.07
0.06 7.5
-3.3
0.05
-3.4
0.04 7
-3.5
0.03
6.5
-3.6 0.02
18.85 18.85 18.85
18.8 18.8 18.8
18.75 18.75 18.75
18.7 18.7 18.7
6.5 7 7.5 -3.6 -3.4 -3.2 0.02 0.04 0.06
T(Gyr) log(Z) E(V-I)
Figure6. (Left)SimulatedSingleStellarPopulationCMDhavinganageof6.9GyrandlogZ=−3.37.Thethick(black)curveshowstheinputisochrone,
withtheoptimalfitisochronerecoveredbythemethodappearingasathin(red)curve.Dottedcurvesshowtheyoungestandoldestisochronesallowedbythe
method,ata1σlevel,whichinfactspana1.0Gyrinterval.(Right) ThesixpanelsshowtheprojectionoftheerrorellipsoidresultingfromtheMonteCarlo
simulationandinversionofaSSPwithage6.9GyrandmetallicitylogZ =−3.37,ontodifferentplanes.Thefilledcirclesshowinputparameters,andthe
emptyonestheaveragevaluesfortherecoveredparameters.
identwhencomparing,forexample,withFigure7.Afewbinaries ever,havingweightedalltheCMDequally,themuchlargermain
canalsobeseenaroundthecriticalturnoffregion,butthesehave sequence population, and the phases beyond the turn off, where
hadlittleeffect.Giventhelargedifferenceinluminositybetween errorsare much smaller, anchor to alarge extent the fit against a
aRGBstarandanymainsequenceone(themostlikelysecondary smallnumber of bluestragglersorbinariesveryclosetotheturn
tobeaddedtoabinary),binarypollutionalongtheRGBproduces off.Morethan2localerrorsigmaawayfromtheturnoff,youcan
absolutelynoeffects. fillinalmostanynumber ofbluestragglers,byconstructiontheir
Bylookingatthebestfitinferredisochronesatthebaseofthe effectwillbenegligible,seeEq.4.
main sequence, it can be seen that these have all shifted towards Inthe context of invertingresolved stellar populations, abi-
theright,inresponsetotheweightingofthedistributionofpoints narydoesnotnecessarilyimplyaphysicalgravitationalassociation
towards this direction. By looking at Figure 8 one can see that betweentwostars,butmerelythatthelightoftwoindividualstars
thisshifttowardstherighthasbeenachievedoptimallybythege- has appeared in thedata as coming from asingle point. The two
neticalgorithmnotbyalteringthereddeningcorrection,butrather starshenceshowasasingleonehavingthetotalluminosityofthe
through an offset in the distance modulus. Whilst inferred age, sumforthetwostars,andaneffectivetemperaturebeingtheenergy
metallicityandreddeningparametershavebeenobtainedwithinthe weightedsumofthetemperaturesofthetwostars,asperaStefan-
1σ errorellipses,theinferreddistancemodulusisoffatarounda Boltzmannlaw.Therefore,especiallyifthinkingofdensesystems
2σlevel,whichhowevercorrespondsonlyto0.15magnitudes.The suchasglobularclusterswherecrowdingeffectscanbedominant,
clearest correlation between theinferredparameters isstillinthe theadoptionofthesameIMFforthesecondarycomponentofthe
age-metallicityplane,andremainsinthesamesenseasexpected. binaryasfortheprimaryoneisjustified.
Otherthantheslightoffsetininferreddistancemodulus,the
onlyothereffectofhavingintroducedarelativelyhighbinarypopu-
4.3 Effectsofdifferentsetsofisochrones
lationhasbeenthebroadeningoftheerrors.Theconfidenceregion
isthistime2.5Gyracross,comparedto1.4GyrforFigure3.Again, Wenowturntoapossiblyimportantsystematic,relatedtotheva-
theisochronesinFigure8areremarkablyclosetoeachother,but lidityof the underlying hypothesis that a single real star from an
althoughregionsbeyondtheturnoffhavenotbeenmuchaffected observed CMD doesinfact behaveand evolveasthestellarevo-
bytheinclusionofthebinaries,thefactthatthemainsequencehas lutionarycodeoneisusingtoconstructanisochronegridassumes
shiftedoutofcorrespondencewiththeseotherregionshasresulted itto.Figure9givestheCMDproducedagainfromthesamepoint
inpoorerfits,andalthoughnosystematicoffsetsinage,metallic- (10.9,−3.05,18.8,0.05) in parameter space, but using this time
ityor reddening correction have appeared, theerror ellipseshave thePadovastellarevolutionarymodels(Girardietal.2002),whilst
certainlyincreasedinsize. inallpreviousexamples,theYaleisochroneswhereused(Demar-
Theparticularcaseofbluestragglershasnotbeentreated,but queetal.2004).Thedistributionofstarsonthemagnitude-colour
a good approximation can be obtained from the binary contami- planeismuchthesameasinFigure3,andnosignificantdifferences
nation example included. Binaries and blue stragglers can popu- canbedetected.
latetheregionclosetotheturnoff,andresultinpoorerfits.How- AnumberofsuchCMDswheretheninvertedusingtheYale